No, the law also prohibited use (without permission) of such things as flash cookies and cookie-like things stored in HTML5's web storage, HTTP ETags, IE userData storage, Silverlight isolated storage, etc. The browser has no control over these things, only standard HTTP cookies for which it is responsible.
The user doesn't have the same fine-grained control. In the case of HTTP cookies you can control whether session cookies are permitted independently of whether persistent cookies are allowed. I believe no such control exists in the domain of local storage.